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Chapter 10: Problem 45

Simplify the expression. $$(4 x y)^{3}$$

Short Answer

Expert verified
64 x^3 y^3

Step by step solution

01

Identify the Expression

We begin with the expression \( (4xy)^3 \)
02

Apply the Power Rule

Apply the power rule \( (a \, b)^n = a^n \, b^n \). This means \( (4xy)^3 = 4^3 \, (xy)^3 \).
03

Simplify the Coefficient

Evaluate \( 4^3 \): \( 4^3 = 64 \).
04

Apply the Exponent to Each Variable

Now distribute the exponent to each variable: \( (xy)^3 = x^3 \, y^3 \).
05

Combine the Results

Combine the simplified coefficient with the simplified variables: \( 64 \, x^3 \, y^3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power rule
The power rule is essential when simplifying expressions that have an exponent. It tells us how to distribute the exponent across the terms inside a parenthesis. For an expression of the form \((a \, b)^n\), the power rule states that we can rewrite this as \(a^n \, b^n\). This rule works because raising a product to a power is the same as raising each factor to that power and then multiplying the results.
Exponent distribution
Exponent distribution is the process of applying an exponent to each term within a parenthesis. Let's look at our example expression \((4xy)^3\). Using the power rule, we first rewrite it as \(4^3 \, (xy)^3\). \[(4xy)^3 = 4^3 \, (xy)^3\]
After this, we proceed by distributing the exponent to each of the individual factors inside the parenthesis. For \((xy)^3\), we distribute the exponent to both \(x\) and \(y\). So, \[(xy)^3 = x^3 \, y^3\].
This gives us \(4^3 \, x^3 \, y^3\).
Simplification
Simplification involves transforming the expression into its simplest form. After distributing the exponent in our example, we now have \((4^3 \, x^3 \, y^3)\). We simplify the constant term first. Calculating \(4^3\), we get \[(4^3 = 64)\]. Next, we combine this with the other terms to get the final simplified expression: \(64 \, x^3 \, y^3\). This process reduces the expression into a form that is easy to understand and work with for future calculations.

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